Home›Calculus 1›Lines and slopes›Problem 13.125

Lines and slopes

Problem 13.125 · easy

Find the slope of the line through \( \displaystyle (-5, 5) \) and \( \displaystyle (3, -5) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.

\[ \frac{-5 - 5}{3 + 5} = - \frac{5}{4} \]
Rise over run.✓ Proved
\[ 5 - \frac{5 \left(x + 5\right)}{4} = - \frac{5 x}{4} - \frac{5}{4} \]
Point-slope form through (-5, 5), then simplify.✓ Proved
\[ \left. - \frac{5 x}{4} - \frac{5}{4} \right|_{\substack{ x=3 }} = -5 \]
It passes through (3, -5) too.✓ Proved

Answer \( m = - \frac{5}{4}\ (\text{decreasing}),\quad y = - \frac{5 x}{4} - \frac{5}{4} \)

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets both points; its slope matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final equation is incorrect. The correct y-intercept is 15/4, not -5/4. The solution incorrectly simplifies 5 - 5(x + 5)/4, likely mishandling the sign of the constant term.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final equation is incorrect; substituting x = -5 into y = -5x/4 - 5/4 yields 5, but the constant term should be 15/4 to satisfy the point (-5, 5). The simplification in step 2 is algebraically wrong (5 - 25/4 = -5/4 is false).
gpt-oss:20b: pass 2026-10-14

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/lines_slopes, checked 2026-10-17 with SymPy 1.14.0.